Show that the Kepler’s law applies to four Galilean moons of Jupiter by calculating the ratio of P 2 A 3 .

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Chapter 3, Problem 37QP

To determine

Show that the Kepler’s law applies to four Galilean moons of Jupiter by calculating the ratio of P2A3.

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(a) Jupiter's third-largest natural satellite, Io, follows an orbit with a semimajor axis of 422,000 km (4.22 ✕ 105 km) and a period of 1.77 Earth days (PIo = 1.77 d). To use Kepler's Third Law, we first must convert Io's orbital semimajor axis to astronomical units. One AU equals 150 million km (1 AU = 1.50 ✕ 108 km). Convert Io's a value to AU and record the result. aIo = AU (b) One Earth year is about 365 days. Convert Io's orbital period to Earth years and record the result. PIo = yr (c) Use the Kepler's Third Law Calculator to calculate Jupiter's mass in solar units. Record the result. MJup(Io) = MSun (d) Based on this result, Jupiter's mass is about that of the Sun. Jupiter has a similar fraction of the Sun's volume. The two objects therefore have rather similar density! In fact, Jupiter has a fairly similar composition as well: most of its mass is in the form of hydrogen and helium.

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